In symmetric spaces(for spacetimes of Einsteinian General Relativity) we would like to find the vector space of Killing vectors($\xi^{(n)}_\mu(x)$) for the given metric tensor($g_{\mu\nu})$ at some fixed point $X$.

Now, for infinitesimal coordinate transformations(of the form $x'^\mu=x^\mu +\epsilon\xi^\mu(x) $) which is really a very special case, we determine the corresponding isometries in terms of the associated Killing vector fields.

This special case yields an interesting result about the *form of the Killing vector field, especially its possible degrees of freedom in terms of initial values of the field and its first order covariant derivative*.

For an infinitesimal patch about a fixed point $X$, the **approximate functional form of the $n$-th Killing vector field** looks like:

$$ \xi^{(n)}_\rho(x)= A^\lambda_\rho(x;X)\xi^{(n)}_\lambda(X)+B^{\lambda\nu}_\rho(x;X)\xi^{(n)}_{\lambda;\nu}(X)$$

Here $A^\lambda_\rho$ and $B^{\lambda\nu}_\rho$ are functions that depend on $g_{\mu\nu}$ and X and are essentially constants for the set of all Killing vector fields about a point $X$.

So, now the argument goes as follows. :

$$\rm No.\,of\, independent\, killing \,vectors=No.\,of\,independent \,parameters\,uniquely\, identifying \,a\,Killing\,vector\,field\\ =N+\dbinom{N}{2}=\dbinom{N+1}{2} $$

Here, $N$=No. of independent initial values $\xi^{(n)}(X)$ and $\dbinom{N}{2}$=No. of independent initial values of covariant derivatives $\xi^{(n)}_{\lambda;\nu}(X)$(because of antisymmetry condition $\xi_{\sigma;\rho}=-\xi_{\rho;\sigma}$)

So, the question is: Is **this approximate calculation(involving Taylor series expansion of Killing vector field components about some fixed point $X$)**

correct?

**It's disturbing to see that approximate methods are being used to yield very concrete answers like maximal number of independent Killing vectors in a vector space.**

*These arguments and line of reasoning are mostly drawn from Weinberg's book on Gravitation and Cosmology*.

## Best Answer

It is not any approximate answer . If you know the value of all derivatives at some point , you can write the function in terms of them . This is not any approximation.